Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,21,22,16)(17,25,27,20)(23,24,28,26)(29,30)(31,36)(32,39)(33,35)(34,41)(37,42)(38,44)(40,43);(7,11)(10,12)(14,19)(15,22)(20,25)(23,28)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,7)(6,10)(8,12)(9,11)(13,15,18,22)(14,16,19,21)(17,23,27,28)(20,24,25,26)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16)(14,15)(17,26)(18,21)(19,22)(20,28)(23,25)(24,27)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(13,17,18,27)(14,20,19,25)(15,23,22,28)(16,24,21,26)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644615);